Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$
Moreover we can consider possibilities of geometric proofs of the following identity for ...

For a Dirichlet character $\chi: \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times} \to \mathbb{C}$, the Dirichlet L function is $$\prod_{q \neq p} (1 - \chi(q)q^{-s})^{-1}$$
If we lift this character to ...

To be clear, I'm having trouble with proving both equalities, and would appreciate a hint. I'm also not sure why $1^+$ must be used as opposed to $1^-$. I'm not sure about the definition of $\zeta(x), ...

I was trying to evaluate the following sum:
$$\sum_{k=0}^{\infty} \frac{1}{(3k+1)^3}$$
W|A gives a nice closed form but I have zero idea about the steps involved to evaluate the sum. How to approach ...

I want to write a program that calculates the number of zeros (It is not necessary to identify them, just the number of them) between 0 and x for the Riemann Zeta function, being x the imaginary part ...

The problem arose, while I was looking at products of power prime zeta functions
$$
P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks},
$$
with $k\in \mathbb{N}$ and $s=it$ with real $t$.
By using ...

I am interested in the following generalization of the Riemann Zeta function:
$$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$
This is most closely related ...

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have:
\begin{equation}
\zeta(2n)=c\pi^{2n},
\end{equation}
where $n$ is a positive integer ...

I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing:
The number of returning paths ...

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as
$$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$
where ...

There are Artin $L$-series and Dirichlet $L$-series, and zeta functions for varieties and for number fields; there are a slew of objects named after Hecke... There are also various kinds of characters ...